競争的および補完的なツール：大規模言語モデルが人間の能力と依存に与える影響の数理モデル

Humans have always externalized thought onto tools, from the tally and the abacus to the map and, now, large language models. I model the agent, the tool, and the task as one dynamical system in which competence (what the user retains) and reliance (what the user outsources) co-evolve, and find that the outcome is bistable. Above a critical tool availability the competent state is destroyed and competence collapses toward a low dependent floor as the user outsources completely. Lowering availability does not reverse the collapse until a far lower threshold, so history of practice rather than the current tool fixes the state. Two users with the same present access can therefore occupy opposite and lasting states, one competent and one dependent, decided only by which they built first. The collapse threshold depends jointly on the competence a user brings to a task and on the tool's transparency, the fraction of its working a user can reconstruct. In the case where an agent faces an uncertain goal, a tool can cause agency itself to transfer to the tool and the human-agent becomes an agentic-instrument, irreversibly, because the tool's model is too large to internalize. The model is tested against several independent data sets, including GPS and map use, arithmetic expertise, and language models. These results reframe how tools should be built, how artificial intelligence is deployed, and what a tool-resistant education might require.